Calculating how long it takes for a diffusion process to effectively reach steady state without computing the transient solution

Calculating how long it takes for a diffusion process to effectively reach steady state without...
Carr, Elliot J.
2017-07-10 00:00:00
Mathematically, it takes an infinite amount of time for the transient solution of a diffusion equation to transition from initial to steady state. Calculating a finite transition time, defined as the time required for the transient solution to transition to within a small prescribed tolerance of the steady-state solution, is much more useful in practice. In this paper, we study estimates of finite transition times that avoid explicit calculation of the transient solution by using the property that the transition to steady state defines a cumulative distribution function when time is treated as a random variable. In total, three approaches are studied: (i) mean action time, (ii) mean plus one standard deviation of action time, and (iii) an approach we derive by approximating the large time asymptotic behavior of the cumulative distribution function. Our approach leads to a simple formula for calculating the finite transition time that depends on the prescribed tolerance δ and the (k−1)th and kth moments (k≥1) of the distribution. Results comparing exact and approximate finite transition times lead to two key findings. First, although the first two approaches are useful at characterizing the time scale of the transition, they do not provide accurate estimates for diffusion processes. Second, the new approach allows one to calculate finite transition times accurate to effectively any number of significant digits using only the moments with the accuracy increasing as the index k is increased.
http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.pngPhysical Review EAmerican Physical Society (APS)http://www.deepdyve.com/lp/american-physical-society-aps/calculating-how-long-it-takes-for-a-diffusion-process-to-effectively-UV3Le0S1QW

Calculating how long it takes for a diffusion process to effectively reach steady state without computing the transient solution

Calculating how long it takes for a diffusion process to effectively reach steady state without computing the transient solution

Abstract

Mathematically, it takes an infinite amount of time for the transient solution of a diffusion equation to transition from initial to steady state. Calculating a finite transition time, defined as the time required for the transient solution to transition to within a small prescribed tolerance of the steady-state solution, is much more useful in practice. In this paper, we study estimates of finite transition times that avoid explicit calculation of the transient solution by using the property that the transition to steady state defines a cumulative distribution function when time is treated as a random variable. In total, three approaches are studied: (i) mean action time, (ii) mean plus one standard deviation of action time, and (iii) an approach we derive by approximating the large time asymptotic behavior of the cumulative distribution function. Our approach leads to a simple formula for calculating the finite transition time that depends on the prescribed tolerance δ and the (k−1)th and kth moments (k≥1) of the distribution. Results comparing exact and approximate finite transition times lead to two key findings. First, although the first two approaches are useful at characterizing the time scale of the transition, they do not provide accurate estimates for diffusion processes. Second, the new approach allows one to calculate finite transition times accurate to effectively any number of significant digits using only the moments with the accuracy increasing as the index k is increased.

Loading next page...

Have problems reading an article?Let us know here.

System error. Please try again!

Thanks for helping us catch any problems with articles on DeepDyve. We'll do our best to fix them.

How was the reading experience on this article?

Check all that apply - Please note that only the first page is available if you have not selected a reading option after clicking "Read Article".

The text was blurry
Page doesn't load Other:

Details

Include any more information that will help us locate the issue and fix it faster for you.

Thank you for submitting a report!

Submitting a report will send us an email through our customer support system.

Abstract

Mathematically, it takes an infinite amount of time for the transient solution of a diffusion equation to transition from initial to steady state. Calculating a finite transition time, defined as the time required for the transient solution to transition to within a small prescribed tolerance of the steady-state solution, is much more useful in practice. In this paper, we study estimates of finite transition times that avoid explicit calculation of the transient solution by using the property that the transition to steady state defines a cumulative distribution function when time is treated as a random variable. In total, three approaches are studied: (i) mean action time, (ii) mean plus one standard deviation of action time, and (iii) an approach we derive by approximating the large time asymptotic behavior of the cumulative distribution function. Our approach leads to a simple formula for calculating the finite transition time that depends on the prescribed tolerance δ and the (k−1)th and kth moments (k≥1) of the distribution. Results comparing exact and approximate finite transition times lead to two key findings. First, although the first two approaches are useful at characterizing the time scale of the transition, they do not provide accurate estimates for diffusion processes. Second, the new approach allows one to calculate finite transition times accurate to effectively any number of significant digits using only the moments with the accuracy increasing as the index k is increased.

Journal

Physical Review E
– American Physical Society (APS)

Published: Jul 10, 2017

Recommended Articles

Loading...

There are no references for this article.

Sorry, we don’t have permission to share this article on DeepDyve,
but here are related articles that you can start reading right now: